Regular components of moduli spaces of stable maps
Author:
Gavril Farkas
Journal:
Proc. Amer. Math. Soc. 131 (2003), 20272036
MSC (2000):
Primary 14H10
Published electronically:
November 4, 2002
MathSciNet review:
1963746
Fulltext PDF Free Access
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Abstract: We construct regular components of the moduli space of stable maps from curves of genus to a product of two projective spaces. These components are generically smooth and have the expected dimension predicted by deformation theory. This result can be seen as a general position theorem for loci in consisting of curves carrying exceptional linear series.
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 E. Arbarello, M. Cornalba, Footnotes to a paper of Beniamino Segre, Math. Annalen 256(1981), 341362. MR 83d:14016
 [ACGH]
 E. Arbarello, M. Cornalba, P. Griffiths, J. Harris, Geometry of algebraic curves, Grundlehren der Mathematischen Wissenschaften, 267, Springer Verlag, 1985. MR 86h:14019
 [BE]
 E. Ballico, Ph. Ellia, On the existence of curves with maximal rank in , J. Reine Angew. Math. 397(1989), 122. MR 90b:14033
 [CGGH]
 J. Carlson, M. Green, P. Griffiths, J. Harris, Infinitesimal variations of Hodge structure (I), Compositio Math. 50(1983), no. 23, 109205. MR 86e:32026a
 [CR]
 M.C. Chang, Z. Ran, Deformations and smoothing of complete linear systems on reducible curves, in: Algebraic geometryBowdoin 1985 (S. Bloch ed.), Proc. Symp. Pure Math. 46, Part 1, 1987, 6375. MR 89f:14020
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 M. Coppens, Some remarks on the schemes , Ann. Mat. Pura Appl. (4) 157(1990), no.1, 183197. MR 92h:14018
 [CKM]
 M. Coppens, C. Keem, G. Martens, The primitive length of a general gonal curve, Indag. Math., N.S., 5(1994), 145159. MR 95c:14021
 [CM]
 M. Coppens, G. Martens, Linear series on a general gonal curve, Abh. Math. Sem. Univ. Hamburg 69(1999), 347371. MR 2000i:14043
 [FP]
 W. Fulton, R. Pandharipande, Notes on stable maps and quantum cohomology, in: Algebraic geometrySanta Cruz 1995 (J. Kollar, R. Lazarsfeld, D. Morrison, eds.), Proc. Symp. Pure Math., 62, Part 2, 1997, 4596. MR 98m:14025
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 J. Harris, D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67(1982), no.1, 2388. MR 83i:14018
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 A. Lopez, On the existence of components of the Hilbert scheme with the expected number of moduli, Comm. Algebra 27(1999), no. 7, 34853493. MR 2001d:14006
 [Ran]
 Z. Ran, Deformations of maps, in: Algebraic Curves and Projective Geometry: Proceedings, Trento 1988, Lecture Notes in Math. 1389, Springer Verlag, 1989, 246253. MR 91f:32021
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 E. Sernesi, On the existence of certain families of curves, Invent. Math. 75(1984), no.1, 2557. MR 85e:14035
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Additional Information
Gavril Farkas
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, Michigan 481091109
Email:
gfarkas@umich.edu
DOI:
http://dx.doi.org/10.1090/S0002993902068144
PII:
S 00029939(02)068144
Received by editor(s):
October 18, 2000
Received by editor(s) in revised form:
February 26, 2002
Published electronically:
November 4, 2002
Communicated by:
Michael Stillman
Article copyright:
© Copyright 2002
American Mathematical Society
